Knowledge of the bulk Lorentz factor Γ0 of gamma-ray bursts (GRBs) allows us to compute their comoving frame properties shedding light on their physics. Upon collisions with the circumburst matter, the fireball of a GRB starts to decelerate, producing a peak or a break (depending on the circumburst density profile) in the light curve of the afterglow. Considering all bursts with known redshift and with an early coverage of their emission, we find 67 GRBs (including one short event) with a peak in their optical or GeV light curves at a time tp. For another 106 GRBs we set an upper limit tpUL. The measure of tp provides the bulk Lorentz factor Γ0 of the fireball before deceleration. We show that tp is due to the dynamics of the fireball deceleration and not to the passage of a characteristic frequency of the synchrotron spectrum across the optical band. Considering the tp of 66 long GRBs and the 85 most constraining upper limits, we estimate Γ0 or a lower limit Γ0LL. Using censored data analysis methods, we reconstruct the most likely distribution of tp. All tp are larger than the time Tp,γ when the prompt γ-ray emission peaks, and are much larger than the time Tph when the fireball becomes transparent, that is, tp>Tp,γ>Tph. The reconstructed distribution of Γ0 has median value ~300 (150) for a uniform (wind) circumburst density profile. In the comoving frame, long GRBs have typical isotropic energy, luminosity, and peak energy ⟨ Eiso ⟩ = 3(8) × 1050 erg, ⟨ Liso ⟩ = 3(15) × 1047 erg s-1, and ⟨ Epeak ⟩ = 1(2) keV in the homogeneous (wind) case. We confirm that the significant correlations between Γ0 and the rest frame isotropic energy (Eiso), luminosity (Liso), and peak energy (Ep) are not due to selection effects. When combined, they lead to the observed Ep−Eiso and Ep−Liso correlations. Finally, assuming a typical opening angle of 5 degrees, we derive the distribution of the jet baryon loading which is centered around a few 10-6M⊙.
Outliers to the peak energy-isotropic energy relation in gamma-ray bursts
